A few years back the state put a bounty on nutria tails. Since the beasts are inedible, even to a cajun, you would drive along the roads of south Louisiana and see dead, tailless rats all over the side of the road. A lot were killed, but they breed prodigiously.
When I visited several years ago, my recollection is that they were having a competition for the best nutria recipe, since as you said they breed prodigiously.
Apparently, they have not caught on as a delicacy.